Part 1: The Infinite Geometric Series 1. Given the following geometric series wr

Question

Part 1: The Infinite Geometric Series 1. Given the following geometric series write out the first 5 partial sums. Explain how to do this. ?_(n=1)^???4(1/3)^n ? S_1 = S_2 = S_3 = S_4 = S_5 2. Do these partial sums appear to be approaching a finite sum for the entire series? If so, what does that sum appear to be? 3. How do we know for sure that the above infinite geometric series has a sum? What is the formula that we can use to find the sum of an infinite geometric series? 4. Identify the following and use them to determine the sum of the infinite geometric series above. r = a_1= Sum = 5. Can we find the sum of these infinite geometric series? In other words, do these series converge? Explain. If you can find the sum, do so. a) ?_(n=1)^???3(-5/2)^n ? b) ?_(n=1)^???1/4 (.125)^n ? Part II: Maclaurin Series Directions: Determine the Maclaurin Series for the following function: f(x)=e^(-3x) for n = 4 1. Determine the first four derivatives of f(x)=e^(-3x). f^I (x)= f^II (x)= f^III (x)= f^IV (x)= 2. Evaluate f(x)=e^(-3x) and the derivatives above at x = 0 f(0)= f^I (0)= f^II (0)= f^III (0)= f^IV (0)= 3. The formula for the Maclaurin Series is f(0)+f^I (0)x+(f^II (0))/2! x^2+f^III/3! x^3+?+(f^n (x))/n! x^n. Plug the values from question 2 into the formula. Then, simplify the fractions and write the Maclaurin Series approximation for f(x)=e^(-3x). 4. Enter f(x)=e^(-3x) and the series approximation from step 3 into a graphing calculator. View the graphs and the table of values. Does the Maclaurin series approximation appear to converge on f(x)=e^(-3x) for all values of x? If so, what does this mean about when I can use the Maclaurin series to approximate f(x)=e^(-3x)? If not, for what values of x can I use the series to approximate f(x)? 5. What would happen to the accuracy of the approximation the Maclaurin Series provides if I increased the number of terms in the series? Why? 6. Determine a Maclaurin Series approximation for f(x)=sin?(6x) where n = 6. Part II: Using MacLaurin Series Maclaurin Series expansions e^x=1+x+x^2/2!+x^3/3!+?+x^n/n! sin?(x)=x-x^3/3!+x^5/5!-

Explanation / Answer

?_(n=1)^???4(1/3)^n ?

above is heometric series

here a = 4/3 and common ratio r = 1/3

so sum = a/(1-r)

= 4/ 3 (2/3)

= 2

above is infinite series

f(x)=e^(-3x)

f ' (x) = -3e^(-3x)

f "(x) = 9e^(-3x)

f ' ' ' (x) = -27e^(-3x)

f ' ' ' ' (x) = 81e^(-3x)

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